Answer:
the length 41.95 feet and the width 20.98 feet
Step-by-step explanation:
We know that "x" and "y" would be the sides of the rectangle.
Therefore we have that the area would be equal to:
x * y = 880
if we solve y, we have to:
y = 880 / x
Now, the cost would be the multiplication of the price by the length of the sides (each side is put twice, since the price goes by the perimeter of the figure):
c = 4 * 2x + 8 * 2y
replacing:
c = 8x + 16 * 880 / x
Now, if we derive with respect to x we are left with:
c '= 8 - 14080 / x ^ 2
if we derive again:
c '' = 28160 / x ^ 3
we calculate the critical point that is when c '= 0
0 = 8 + 14080 / x ^ 2
x ^ 2 = 14080/8
x ^ 2 = 1760
x = + -41.95
Now for x = 41.95, we evaluate in the second derivative to know if it is a minimum or maximum value:
28160 / 41.95 ^ 3 = 0.381, is a number greater than 0, therefore it is a minimum value, therefore it is value is correct, now for y:
y = 880 / 41.95 = 20.98
which means that for it to be cheaper the length must be 41.95 feet and the width 20.98 feet
